Over a million tosses you'll have almost the same amount
of heads as tails

Well, yes, and no. It depends on how you look at it.

After a million tosses of a fair coin, you can expect that
the numbers of heads and tails will differ by about 1,000. This is a
pretty big number.

On the other hand, after a million tosses of a fair coin, you can
expect that the numbers of heads and tails will differ by about 0.1%.
This is a pretty small number.

In general, if you flip the coin n times, the expected
difference between the numbers of heads and tails will be about
√n. As n gets larger, so does √n.
So the more times you flip the coin, the larger the expected
difference in the two totals.

But the relative difference is the quotient of the difference and the
total number of flips; that is, √n/n =
1/√n. As n gets larger, 1/√n goes to
zero. So the more times you flip the coin, the smaller the expected difference
in the two totals.

It's not quite right to say that you will have "almost the same
amount of heads as tails". But it's not quite wrong either. As you
flip the coin more and more, you can expect the totals to get farther
and farther apart—but the difference between them will be less and
less significant, compared with the totals themselves.

[ Addendum 20060720: Although the main point of this article is
correct, I made some specific technical errors. A correction is available. ]